/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.koat /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^2), O(n^2)) proof of /export/starexec/sandbox/benchmark/theBenchmark.koat # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, n^2). (0) CpxIntTrs (1) Koat Proof [FINISHED, 293 ms] (2) BOUNDS(1, n^2) (3) Loat Proof [FINISHED, 906 ms] (4) BOUNDS(n^2, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_speedSimpleMultipleDep_start(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_speedSimpleMultipleDep_bb0_in(v_m, v_n, v_x_0, v_y_0)) :|: TRUE eval_speedSimpleMultipleDep_bb0_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_speedSimpleMultipleDep_0(v_m, v_n, v_x_0, v_y_0)) :|: TRUE eval_speedSimpleMultipleDep_0(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_speedSimpleMultipleDep_1(v_m, v_n, v_x_0, v_y_0)) :|: TRUE eval_speedSimpleMultipleDep_1(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_speedSimpleMultipleDep_2(v_m, v_n, v_x_0, v_y_0)) :|: TRUE eval_speedSimpleMultipleDep_2(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_speedSimpleMultipleDep_3(v_m, v_n, v_x_0, v_y_0)) :|: TRUE eval_speedSimpleMultipleDep_3(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_speedSimpleMultipleDep_4(v_m, v_n, v_x_0, v_y_0)) :|: TRUE eval_speedSimpleMultipleDep_4(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_speedSimpleMultipleDep_5(v_m, v_n, v_x_0, v_y_0)) :|: TRUE eval_speedSimpleMultipleDep_5(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_speedSimpleMultipleDep_6(v_m, v_n, v_x_0, v_y_0)) :|: TRUE eval_speedSimpleMultipleDep_6(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_speedSimpleMultipleDep_7(v_m, v_n, v_x_0, v_y_0)) :|: TRUE eval_speedSimpleMultipleDep_7(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_speedSimpleMultipleDep_bb1_in(v_m, v_n, 0, 0)) :|: TRUE eval_speedSimpleMultipleDep_bb1_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_speedSimpleMultipleDep_bb2_in(v_m, v_n, v_x_0, v_y_0)) :|: v_x_0 < v_n eval_speedSimpleMultipleDep_bb1_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_speedSimpleMultipleDep_bb3_in(v_m, v_n, v_x_0, v_y_0)) :|: v_x_0 >= v_n eval_speedSimpleMultipleDep_bb2_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_speedSimpleMultipleDep_bb1_in(v_m, v_n, v_x_0, v_y_0 + 1)) :|: v_y_0 < v_m eval_speedSimpleMultipleDep_bb2_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_speedSimpleMultipleDep_bb1_in(v_m, v_n, v_x_0 + 1, v_y_0 + 1)) :|: v_y_0 < v_m && v_y_0 >= v_m eval_speedSimpleMultipleDep_bb2_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_speedSimpleMultipleDep_bb1_in(v_m, v_n, v_x_0, 0)) :|: v_y_0 >= v_m && v_y_0 < v_m eval_speedSimpleMultipleDep_bb2_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_speedSimpleMultipleDep_bb1_in(v_m, v_n, v_x_0 + 1, 0)) :|: v_y_0 >= v_m eval_speedSimpleMultipleDep_bb3_in(v_m, v_n, v_x_0, v_y_0) -> Com_1(eval_speedSimpleMultipleDep_stop(v_m, v_n, v_x_0, v_y_0)) :|: TRUE The start-symbols are:[eval_speedSimpleMultipleDep_start_4] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 2*ar_2 + 2*ar_2*ar_3 + 2*ar_3 + 15) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb0in(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep0(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDep0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep1(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDep1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep2(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDep2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep3(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDep3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep4(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDep4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep5(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDep5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep6(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDep6(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep7(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDep7(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(0, 0, ar_2, ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_3 >= ar_1 + 1 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0 + 1, ar_1 + 1, ar_2, ar_3)) [ ar_3 >= ar_1 + 1 /\ ar_1 >= ar_3 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0, 0, ar_2, ar_3)) [ ar_1 >= ar_3 /\ ar_3 >= ar_1 + 1 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0 + 1, 0, ar_2, ar_3)) [ ar_1 >= ar_3 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepstop(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Testing for reachability in the complexity graph removes the following transitions from problem 1: evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0 + 1, ar_1 + 1, ar_2, ar_3)) [ ar_3 >= ar_1 + 1 /\ ar_1 >= ar_3 ] evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0, 0, ar_2, ar_3)) [ ar_1 >= ar_3 /\ ar_3 >= ar_1 + 1 ] We thus obtain the following problem: 2: T: (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepstop(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0 + 1, 0, ar_2, ar_3)) [ ar_1 >= ar_3 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_3 >= ar_1 + 1 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDep7(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(0, 0, ar_2, ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDep6(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep7(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDep5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep6(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDep4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep5(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDep3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep4(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDep2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep3(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDep1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep2(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDep0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep1(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep0(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb0in(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 2 produces the following problem: 3: T: (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepstop(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0 + 1, 0, ar_2, ar_3)) [ ar_1 >= ar_3 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_3 >= ar_1 + 1 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ] (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep7(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(0, 0, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep6(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep7(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep6(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep5(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep4(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep3(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep2(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep1(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDepbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep0(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDepstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb0in(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalspeedSimpleMultipleDepbb3in) = 1 Pol(evalspeedSimpleMultipleDepstop) = 0 Pol(evalspeedSimpleMultipleDepbb2in) = 2 Pol(evalspeedSimpleMultipleDepbb1in) = 2 Pol(evalspeedSimpleMultipleDep7) = 2 Pol(evalspeedSimpleMultipleDep6) = 2 Pol(evalspeedSimpleMultipleDep5) = 2 Pol(evalspeedSimpleMultipleDep4) = 2 Pol(evalspeedSimpleMultipleDep3) = 2 Pol(evalspeedSimpleMultipleDep2) = 2 Pol(evalspeedSimpleMultipleDep1) = 2 Pol(evalspeedSimpleMultipleDep0) = 2 Pol(evalspeedSimpleMultipleDepbb0in) = 2 Pol(evalspeedSimpleMultipleDepstart) = 2 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalspeedSimpleMultipleDepbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepstop(ar_0, ar_1, ar_2, ar_3)) evalspeedSimpleMultipleDepbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ] strictly and produces the following problem: 4: T: (Comp: 2, Cost: 1) evalspeedSimpleMultipleDepbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepstop(ar_0, ar_1, ar_2, ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0 + 1, 0, ar_2, ar_3)) [ ar_1 >= ar_3 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_3 >= ar_1 + 1 ] (Comp: 2, Cost: 1) evalspeedSimpleMultipleDepbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_0 >= ar_2 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_2 >= ar_0 + 1 ] (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep7(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(0, 0, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep6(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep7(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep6(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep5(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep4(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep3(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep2(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep1(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDepbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep0(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDepstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb0in(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Applied AI with 'oct' on problem 4 to obtain the following invariants: For symbol evalspeedSimpleMultipleDepbb1in: X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0 For symbol evalspeedSimpleMultipleDepbb2in: X_3 - 1 >= 0 /\ X_2 + X_3 - 1 >= 0 /\ X_1 + X_3 - 1 >= 0 /\ -X_1 + X_3 - 1 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0 For symbol evalspeedSimpleMultipleDepbb3in: X_1 - X_3 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0 This yielded the following problem: 5: T: (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] (Comp: 1, Cost: 1) evalspeedSimpleMultipleDepstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb0in(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDepbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep0(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep1(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep2(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep3(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep4(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep5(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep6(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep6(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep7(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep7(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(0, 0, ar_2, ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_2 >= ar_0 + 1 ] (Comp: 2, Cost: 1) evalspeedSimpleMultipleDepbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_0 >= ar_2 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_3 >= ar_1 + 1 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0 + 1, 0, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_1 >= ar_3 ] (Comp: 2, Cost: 1) evalspeedSimpleMultipleDepbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepstop(ar_0, ar_1, ar_2, ar_3)) [ ar_0 - ar_2 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(koat_start) = V_3 Pol(evalspeedSimpleMultipleDepstart) = V_3 Pol(evalspeedSimpleMultipleDepbb0in) = V_3 Pol(evalspeedSimpleMultipleDep0) = V_3 Pol(evalspeedSimpleMultipleDep1) = V_3 Pol(evalspeedSimpleMultipleDep2) = V_3 Pol(evalspeedSimpleMultipleDep3) = V_3 Pol(evalspeedSimpleMultipleDep4) = V_3 Pol(evalspeedSimpleMultipleDep5) = V_3 Pol(evalspeedSimpleMultipleDep6) = V_3 Pol(evalspeedSimpleMultipleDep7) = V_3 Pol(evalspeedSimpleMultipleDepbb1in) = -V_1 + V_3 Pol(evalspeedSimpleMultipleDepbb2in) = -V_1 + V_3 Pol(evalspeedSimpleMultipleDepbb3in) = -V_1 + V_3 Pol(evalspeedSimpleMultipleDepstop) = -V_1 + V_3 orients all transitions weakly and the transition evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0 + 1, 0, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_1 >= ar_3 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] (Comp: 1, Cost: 1) evalspeedSimpleMultipleDepstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb0in(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDepbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep0(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep1(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep2(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep3(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep4(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep5(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep6(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep6(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep7(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep7(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(0, 0, ar_2, ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_2 >= ar_0 + 1 ] (Comp: 2, Cost: 1) evalspeedSimpleMultipleDepbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_0 >= ar_2 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_3 >= ar_1 + 1 ] (Comp: ar_2, Cost: 1) evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0 + 1, 0, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_1 >= ar_3 ] (Comp: 2, Cost: 1) evalspeedSimpleMultipleDepbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepstop(ar_0, ar_1, ar_2, ar_3)) [ ar_0 - ar_2 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalspeedSimpleMultipleDepbb2in) = -V_2 + V_4 Pol(evalspeedSimpleMultipleDepbb1in) = -V_2 + V_4 and size complexities S("evalspeedSimpleMultipleDepbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepstop(ar_0, ar_1, ar_2, ar_3)) [ ar_0 - ar_2 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 ]", 0-0) = ar_2 S("evalspeedSimpleMultipleDepbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepstop(ar_0, ar_1, ar_2, ar_3)) [ ar_0 - ar_2 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 ]", 0-1) = 0 S("evalspeedSimpleMultipleDepbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepstop(ar_0, ar_1, ar_2, ar_3)) [ ar_0 - ar_2 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 ]", 0-2) = ar_2 S("evalspeedSimpleMultipleDepbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepstop(ar_0, ar_1, ar_2, ar_3)) [ ar_0 - ar_2 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 ]", 0-3) = ar_3 S("evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0 + 1, 0, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 1 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ -ar_0 + ar_2 - 1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_1 >= ar_3 ]", 0-0) = ar_2 S("evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0 + 1, 0, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 1 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ -ar_0 + ar_2 - 1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_1 >= ar_3 ]", 0-1) = 0 S("evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0 + 1, 0, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 1 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ -ar_0 + ar_2 - 1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_1 >= ar_3 ]", 0-2) = ar_2 S("evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0 + 1, 0, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 1 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ -ar_0 + ar_2 - 1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_1 >= ar_3 ]", 0-3) = ar_3 S("evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 1 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ -ar_0 + ar_2 - 1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_3 >= ar_1 + 1 ]", 0-0) = ar_2 S("evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 1 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ -ar_0 + ar_2 - 1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_3 >= ar_1 + 1 ]", 0-1) = ar_3 S("evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 1 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ -ar_0 + ar_2 - 1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_3 >= ar_1 + 1 ]", 0-2) = ar_2 S("evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 1 >= 0 /\\ ar_0 + ar_2 - 1 >= 0 /\\ -ar_0 + ar_2 - 1 >= 0 /\\ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_3 >= ar_1 + 1 ]", 0-3) = ar_3 S("evalspeedSimpleMultipleDepbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_0 >= ar_2 ]", 0-0) = ar_2 S("evalspeedSimpleMultipleDepbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_0 >= ar_2 ]", 0-1) = 0 S("evalspeedSimpleMultipleDepbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_0 >= ar_2 ]", 0-2) = ar_2 S("evalspeedSimpleMultipleDepbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_0 >= ar_2 ]", 0-3) = ar_3 S("evalspeedSimpleMultipleDepbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_2 >= ar_0 + 1 ]", 0-0) = ar_2 S("evalspeedSimpleMultipleDepbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_2 >= ar_0 + 1 ]", 0-1) = ar_3 S("evalspeedSimpleMultipleDepbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_2 >= ar_0 + 1 ]", 0-2) = ar_2 S("evalspeedSimpleMultipleDepbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\\ ar_0 + ar_1 >= 0 /\\ ar_0 >= 0 /\\ ar_2 >= ar_0 + 1 ]", 0-3) = ar_3 S("evalspeedSimpleMultipleDep7(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(0, 0, ar_2, ar_3))", 0-0) = 0 S("evalspeedSimpleMultipleDep7(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(0, 0, ar_2, ar_3))", 0-1) = 0 S("evalspeedSimpleMultipleDep7(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(0, 0, ar_2, ar_3))", 0-2) = ar_2 S("evalspeedSimpleMultipleDep7(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(0, 0, ar_2, ar_3))", 0-3) = ar_3 S("evalspeedSimpleMultipleDep6(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep7(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 S("evalspeedSimpleMultipleDep6(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep7(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalspeedSimpleMultipleDep6(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep7(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalspeedSimpleMultipleDep6(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep7(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("evalspeedSimpleMultipleDep5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep6(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 S("evalspeedSimpleMultipleDep5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep6(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalspeedSimpleMultipleDep5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep6(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalspeedSimpleMultipleDep5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep6(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("evalspeedSimpleMultipleDep4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep5(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 S("evalspeedSimpleMultipleDep4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep5(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalspeedSimpleMultipleDep4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep5(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalspeedSimpleMultipleDep4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep5(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("evalspeedSimpleMultipleDep3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep4(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 S("evalspeedSimpleMultipleDep3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep4(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalspeedSimpleMultipleDep3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep4(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalspeedSimpleMultipleDep3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep4(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("evalspeedSimpleMultipleDep2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep3(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 S("evalspeedSimpleMultipleDep2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep3(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalspeedSimpleMultipleDep2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep3(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalspeedSimpleMultipleDep2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep3(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("evalspeedSimpleMultipleDep1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep2(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 S("evalspeedSimpleMultipleDep1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep2(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalspeedSimpleMultipleDep1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep2(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalspeedSimpleMultipleDep1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep2(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("evalspeedSimpleMultipleDep0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep1(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 S("evalspeedSimpleMultipleDep0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep1(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalspeedSimpleMultipleDep0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep1(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalspeedSimpleMultipleDep0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep1(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("evalspeedSimpleMultipleDepbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep0(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 S("evalspeedSimpleMultipleDepbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep0(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalspeedSimpleMultipleDepbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep0(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalspeedSimpleMultipleDepbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep0(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("evalspeedSimpleMultipleDepstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb0in(ar_0, ar_1, ar_2, ar_3))", 0-0) = ar_0 S("evalspeedSimpleMultipleDepstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb0in(ar_0, ar_1, ar_2, ar_3))", 0-1) = ar_1 S("evalspeedSimpleMultipleDepstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb0in(ar_0, ar_1, ar_2, ar_3))", 0-2) = ar_2 S("evalspeedSimpleMultipleDepstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb0in(ar_0, ar_1, ar_2, ar_3))", 0-3) = ar_3 S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-0) = ar_0 S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-1) = ar_1 S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-2) = ar_2 S("koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ]", 0-3) = ar_3 orients the transitions evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_3 >= ar_1 + 1 ] evalspeedSimpleMultipleDepbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_2 >= ar_0 + 1 ] weakly and the transition evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_3 >= ar_1 + 1 ] strictly and produces the following problem: 7: T: (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] (Comp: 1, Cost: 1) evalspeedSimpleMultipleDepstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb0in(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDepbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep0(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep1(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep2(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep3(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep4(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep5(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep6(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep6(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep7(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep7(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(0, 0, ar_2, ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_2 >= ar_0 + 1 ] (Comp: 2, Cost: 1) evalspeedSimpleMultipleDepbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_0 >= ar_2 ] (Comp: ar_2*ar_3 + ar_3, Cost: 1) evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_3 >= ar_1 + 1 ] (Comp: ar_2, Cost: 1) evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0 + 1, 0, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_1 >= ar_3 ] (Comp: 2, Cost: 1) evalspeedSimpleMultipleDepbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepstop(ar_0, ar_1, ar_2, ar_3)) [ ar_0 - ar_2 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 7 produces the following problem: 8: T: (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepstart(ar_0, ar_1, ar_2, ar_3)) [ 0 <= 0 ] (Comp: 1, Cost: 1) evalspeedSimpleMultipleDepstart(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb0in(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDepbb0in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep0(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep0(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep1(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep1(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep2(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep2(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep3(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep3(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep4(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep4(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep5(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep5(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep6(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep6(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDep7(ar_0, ar_1, ar_2, ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDep7(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(0, 0, ar_2, ar_3)) (Comp: ar_2 + ar_2*ar_3 + ar_3 + 1, Cost: 1) evalspeedSimpleMultipleDepbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_2 >= ar_0 + 1 ] (Comp: 2, Cost: 1) evalspeedSimpleMultipleDepbb1in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb3in(ar_0, ar_1, ar_2, ar_3)) [ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_0 >= ar_2 ] (Comp: ar_2*ar_3 + ar_3, Cost: 1) evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0, ar_1 + 1, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_3 >= ar_1 + 1 ] (Comp: ar_2, Cost: 1) evalspeedSimpleMultipleDepbb2in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(ar_0 + 1, 0, ar_2, ar_3)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 1 >= 0 /\ ar_0 + ar_2 - 1 >= 0 /\ -ar_0 + ar_2 - 1 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 /\ ar_1 >= ar_3 ] (Comp: 2, Cost: 1) evalspeedSimpleMultipleDepbb3in(ar_0, ar_1, ar_2, ar_3) -> Com_1(evalspeedSimpleMultipleDepstop(ar_0, ar_1, ar_2, ar_3)) [ ar_0 - ar_2 >= 0 /\ ar_1 >= 0 /\ ar_0 + ar_1 >= 0 /\ ar_0 >= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 2*ar_2 + 2*ar_2*ar_3 + 2*ar_3 + 15 Time: 0.377 sec (SMT: 0.307 sec) ---------------------------------------- (2) BOUNDS(1, n^2) ---------------------------------------- (3) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: evalspeedSimpleMultipleDepstart 0: evalspeedSimpleMultipleDepstart -> evalspeedSimpleMultipleDepbb0in : [], cost: 1 1: evalspeedSimpleMultipleDepbb0in -> evalspeedSimpleMultipleDep0 : [], cost: 1 2: evalspeedSimpleMultipleDep0 -> evalspeedSimpleMultipleDep1 : [], cost: 1 3: evalspeedSimpleMultipleDep1 -> evalspeedSimpleMultipleDep2 : [], cost: 1 4: evalspeedSimpleMultipleDep2 -> evalspeedSimpleMultipleDep3 : [], cost: 1 5: evalspeedSimpleMultipleDep3 -> evalspeedSimpleMultipleDep4 : [], cost: 1 6: evalspeedSimpleMultipleDep4 -> evalspeedSimpleMultipleDep5 : [], cost: 1 7: evalspeedSimpleMultipleDep5 -> evalspeedSimpleMultipleDep6 : [], cost: 1 8: evalspeedSimpleMultipleDep6 -> evalspeedSimpleMultipleDep7 : [], cost: 1 9: evalspeedSimpleMultipleDep7 -> evalspeedSimpleMultipleDepbb1in : A'=0, B'=0, [], cost: 1 10: evalspeedSimpleMultipleDepbb1in -> evalspeedSimpleMultipleDepbb2in : [ C>=1+A ], cost: 1 11: evalspeedSimpleMultipleDepbb1in -> evalspeedSimpleMultipleDepbb3in : [ A>=C ], cost: 1 12: evalspeedSimpleMultipleDepbb2in -> evalspeedSimpleMultipleDepbb1in : B'=1+B, [ D>=1+B ], cost: 1 13: evalspeedSimpleMultipleDepbb2in -> evalspeedSimpleMultipleDepbb1in : A'=1+A, B'=1+B, [ D>=1+B && B>=D ], cost: 1 14: evalspeedSimpleMultipleDepbb2in -> evalspeedSimpleMultipleDepbb1in : B'=0, [ B>=D && D>=1+B ], cost: 1 15: evalspeedSimpleMultipleDepbb2in -> evalspeedSimpleMultipleDepbb1in : A'=1+A, B'=0, [ B>=D ], cost: 1 16: evalspeedSimpleMultipleDepbb3in -> evalspeedSimpleMultipleDepstop : [], cost: 1 Removed unreachable and leaf rules: Start location: evalspeedSimpleMultipleDepstart 0: evalspeedSimpleMultipleDepstart -> evalspeedSimpleMultipleDepbb0in : [], cost: 1 1: evalspeedSimpleMultipleDepbb0in -> evalspeedSimpleMultipleDep0 : [], cost: 1 2: evalspeedSimpleMultipleDep0 -> evalspeedSimpleMultipleDep1 : [], cost: 1 3: evalspeedSimpleMultipleDep1 -> evalspeedSimpleMultipleDep2 : [], cost: 1 4: evalspeedSimpleMultipleDep2 -> evalspeedSimpleMultipleDep3 : [], cost: 1 5: evalspeedSimpleMultipleDep3 -> evalspeedSimpleMultipleDep4 : [], cost: 1 6: evalspeedSimpleMultipleDep4 -> evalspeedSimpleMultipleDep5 : [], cost: 1 7: evalspeedSimpleMultipleDep5 -> evalspeedSimpleMultipleDep6 : [], cost: 1 8: evalspeedSimpleMultipleDep6 -> evalspeedSimpleMultipleDep7 : [], cost: 1 9: evalspeedSimpleMultipleDep7 -> evalspeedSimpleMultipleDepbb1in : A'=0, B'=0, [], cost: 1 10: evalspeedSimpleMultipleDepbb1in -> evalspeedSimpleMultipleDepbb2in : [ C>=1+A ], cost: 1 12: evalspeedSimpleMultipleDepbb2in -> evalspeedSimpleMultipleDepbb1in : B'=1+B, [ D>=1+B ], cost: 1 13: evalspeedSimpleMultipleDepbb2in -> evalspeedSimpleMultipleDepbb1in : A'=1+A, B'=1+B, [ D>=1+B && B>=D ], cost: 1 14: evalspeedSimpleMultipleDepbb2in -> evalspeedSimpleMultipleDepbb1in : B'=0, [ B>=D && D>=1+B ], cost: 1 15: evalspeedSimpleMultipleDepbb2in -> evalspeedSimpleMultipleDepbb1in : A'=1+A, B'=0, [ B>=D ], cost: 1 Removed rules with unsatisfiable guard: Start location: evalspeedSimpleMultipleDepstart 0: evalspeedSimpleMultipleDepstart -> evalspeedSimpleMultipleDepbb0in : [], cost: 1 1: evalspeedSimpleMultipleDepbb0in -> evalspeedSimpleMultipleDep0 : [], cost: 1 2: evalspeedSimpleMultipleDep0 -> evalspeedSimpleMultipleDep1 : [], cost: 1 3: evalspeedSimpleMultipleDep1 -> evalspeedSimpleMultipleDep2 : [], cost: 1 4: evalspeedSimpleMultipleDep2 -> evalspeedSimpleMultipleDep3 : [], cost: 1 5: evalspeedSimpleMultipleDep3 -> evalspeedSimpleMultipleDep4 : [], cost: 1 6: evalspeedSimpleMultipleDep4 -> evalspeedSimpleMultipleDep5 : [], cost: 1 7: evalspeedSimpleMultipleDep5 -> evalspeedSimpleMultipleDep6 : [], cost: 1 8: evalspeedSimpleMultipleDep6 -> evalspeedSimpleMultipleDep7 : [], cost: 1 9: evalspeedSimpleMultipleDep7 -> evalspeedSimpleMultipleDepbb1in : A'=0, B'=0, [], cost: 1 10: evalspeedSimpleMultipleDepbb1in -> evalspeedSimpleMultipleDepbb2in : [ C>=1+A ], cost: 1 12: evalspeedSimpleMultipleDepbb2in -> evalspeedSimpleMultipleDepbb1in : B'=1+B, [ D>=1+B ], cost: 1 15: evalspeedSimpleMultipleDepbb2in -> evalspeedSimpleMultipleDepbb1in : A'=1+A, B'=0, [ B>=D ], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: evalspeedSimpleMultipleDepstart 25: evalspeedSimpleMultipleDepstart -> evalspeedSimpleMultipleDepbb1in : A'=0, B'=0, [], cost: 10 10: evalspeedSimpleMultipleDepbb1in -> evalspeedSimpleMultipleDepbb2in : [ C>=1+A ], cost: 1 12: evalspeedSimpleMultipleDepbb2in -> evalspeedSimpleMultipleDepbb1in : B'=1+B, [ D>=1+B ], cost: 1 15: evalspeedSimpleMultipleDepbb2in -> evalspeedSimpleMultipleDepbb1in : A'=1+A, B'=0, [ B>=D ], cost: 1 Eliminated locations (on tree-shaped paths): Start location: evalspeedSimpleMultipleDepstart 25: evalspeedSimpleMultipleDepstart -> evalspeedSimpleMultipleDepbb1in : A'=0, B'=0, [], cost: 10 26: evalspeedSimpleMultipleDepbb1in -> evalspeedSimpleMultipleDepbb1in : B'=1+B, [ C>=1+A && D>=1+B ], cost: 2 27: evalspeedSimpleMultipleDepbb1in -> evalspeedSimpleMultipleDepbb1in : A'=1+A, B'=0, [ C>=1+A && B>=D ], cost: 2 Accelerating simple loops of location 10. Accelerating the following rules: 26: evalspeedSimpleMultipleDepbb1in -> evalspeedSimpleMultipleDepbb1in : B'=1+B, [ C>=1+A && D>=1+B ], cost: 2 27: evalspeedSimpleMultipleDepbb1in -> evalspeedSimpleMultipleDepbb1in : A'=1+A, B'=0, [ C>=1+A && B>=D ], cost: 2 Accelerated rule 26 with metering function D-B, yielding the new rule 28. Accelerated rule 27 with metering function C-A (after strengthening guard), yielding the new rule 29. Nested simple loops 27 (outer loop) and 28 (inner loop) with metering function -1+C-A, resulting in the new rules: 30, 31. Removing the simple loops: 26 27. Accelerated all simple loops using metering functions (where possible): Start location: evalspeedSimpleMultipleDepstart 25: evalspeedSimpleMultipleDepstart -> evalspeedSimpleMultipleDepbb1in : A'=0, B'=0, [], cost: 10 28: evalspeedSimpleMultipleDepbb1in -> evalspeedSimpleMultipleDepbb1in : B'=D, [ C>=1+A && D>=1+B ], cost: 2*D-2*B 29: evalspeedSimpleMultipleDepbb1in -> evalspeedSimpleMultipleDepbb1in : A'=C, B'=0, [ C>=1+A && B>=D && 0>=D ], cost: 2*C-2*A 30: evalspeedSimpleMultipleDepbb1in -> evalspeedSimpleMultipleDepbb1in : A'=-1+C, B'=D, [ B>=D && C>=2+A && D>=1 ], cost: -2+2*C+2*(-1+C-A)*D-2*A 31: evalspeedSimpleMultipleDepbb1in -> evalspeedSimpleMultipleDepbb1in : A'=-1+C, B'=D, [ D>=1+B && C>=2+A && D>=1 ], cost: -2+2*C+2*(-1+C-A)*D+2*D-2*A-2*B Chained accelerated rules (with incoming rules): Start location: evalspeedSimpleMultipleDepstart 25: evalspeedSimpleMultipleDepstart -> evalspeedSimpleMultipleDepbb1in : A'=0, B'=0, [], cost: 10 32: evalspeedSimpleMultipleDepstart -> evalspeedSimpleMultipleDepbb1in : A'=0, B'=D, [ C>=1 && D>=1 ], cost: 10+2*D 33: evalspeedSimpleMultipleDepstart -> evalspeedSimpleMultipleDepbb1in : A'=C, B'=0, [ C>=1 && 0>=D ], cost: 10+2*C 34: evalspeedSimpleMultipleDepstart -> evalspeedSimpleMultipleDepbb1in : A'=-1+C, B'=D, [ D>=1 && C>=2 ], cost: 8+2*D*(-1+C)+2*C+2*D Removed unreachable locations (and leaf rules with constant cost): Start location: evalspeedSimpleMultipleDepstart 32: evalspeedSimpleMultipleDepstart -> evalspeedSimpleMultipleDepbb1in : A'=0, B'=D, [ C>=1 && D>=1 ], cost: 10+2*D 33: evalspeedSimpleMultipleDepstart -> evalspeedSimpleMultipleDepbb1in : A'=C, B'=0, [ C>=1 && 0>=D ], cost: 10+2*C 34: evalspeedSimpleMultipleDepstart -> evalspeedSimpleMultipleDepbb1in : A'=-1+C, B'=D, [ D>=1 && C>=2 ], cost: 8+2*D*(-1+C)+2*C+2*D ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evalspeedSimpleMultipleDepstart 32: evalspeedSimpleMultipleDepstart -> evalspeedSimpleMultipleDepbb1in : A'=0, B'=D, [ C>=1 && D>=1 ], cost: 10+2*D 33: evalspeedSimpleMultipleDepstart -> evalspeedSimpleMultipleDepbb1in : A'=C, B'=0, [ C>=1 && 0>=D ], cost: 10+2*C 34: evalspeedSimpleMultipleDepstart -> evalspeedSimpleMultipleDepbb1in : A'=-1+C, B'=D, [ D>=1 && C>=2 ], cost: 8+2*D*(-1+C)+2*C+2*D Computing asymptotic complexity for rule 32 Solved the limit problem by the following transformations: Created initial limit problem: 10+2*D (+), C (+/+!), D (+/+!) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {C==n,D==n} resulting limit problem: [solved] Solution: C / n D / n Resulting cost 10+2*n has complexity: Poly(n^1) Found new complexity Poly(n^1). Computing asymptotic complexity for rule 34 Solved the limit problem by the following transformations: Created initial limit problem: D (+/+!), -1+C (+/+!), 8+2*C+2*C*D (+) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {C==n,D==n} resulting limit problem: [solved] Solution: C / n D / n Resulting cost 8+2*n^2+2*n has complexity: Poly(n^2) Found new complexity Poly(n^2). Obtained the following overall complexity (w.r.t. the length of the input n): Complexity: Poly(n^2) Cpx degree: 2 Solved cost: 8+2*n^2+2*n Rule cost: 8+2*D*(-1+C)+2*C+2*D Rule guard: [ D>=1 && C>=2 ] WORST_CASE(Omega(n^2),?) ---------------------------------------- (4) BOUNDS(n^2, INF)